Dear Forest!
You wrote:
> Let x, y, and z be positive integers such that x+y+z=N, and max(x,y,z)<N/2,
> where N is the number of some large population of voters, and the ordinal
> preferences are divided into three factions: x: A>B>C y: B>C>A z: C>A>B
> Further assume that the cardinal ratings of the middle candidate within each
> faction are distributed uniformly, so that in the first faction the cardinal
> ratings of B are distributed evenly between zero and 100%. Let (alpha,
> beta, gamma) be a "lottery" for this election. Then the number of voters
> that prefer A to this lottery is given by the expression p(A) = x +
> beta*z/(gamma+beta) Corresponding expressions for B and C are p(B)
> = y + gamma*x/(alpha+gamma) and p(C) = z + alpha*y/(beta+alpha) If
> we set (alpha, beta, gamma) equal to (x+y-z, y+z-x, z+x-y)/(2*N) ,
> then p(A)=p(B)=p(C)=N/2 , which means that none of the candidates is
> preferred over the lottery by more than half of the population. Isn't that
> inter!
esting? Forest
It is interesting indeed. It also holds when voters judge by median utility
instead of mean utility (and then we don't need the uniform distribution
assumption)...
Jobst
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